120A Math Introduction to Group Theory: Question sheet 7
Hand in questions 6, 8(c) & 9 at lecture Friday 22nd May 1. (a) Define the notions of kernel and image of a homomorphism : G H of groups. (b) Prove that the kernel of a homomorphism : G H is a normal subgroup of G (Recall that g1 g ker g-1 g1 ker ). (c) Give an example to show that the image of need not be normal in H? (Hint: only nonAbelian groups have non-normal subgroups...). 2. Decide whether each of the following maps is a homomorphism. For those that are, find the kernel. (a) : (R , ) (R , ) where ( x ) = | x |. (b) : Z4 S3 where (n) = (12)n . (c) : Z4 S3 where (n) = (123)n . (d) : (C, +) (C , ) where (z) = e2
-1 z .
3. In this question we find all the automorphisms : Zn Zn . (a) Let a be constant in Zn , and consider the function : Zn Zn defined by ( x ) = ax (mod n). Show that is well-defined and a homomorphism Zn Zn . (b) Why are all homomorphisms : Zn Zn of the form given in part (a). (c) Find all the automorphisms of Z4 , Z5 , and Z6 . (d) Given n, find all the automorphisms : Zn Zn . Prove your assertion. 4. Consider the map : G G given by ( g) = g-1 . For what groups G is a homomorphism? An automorphism? Prove your assertions. 5. Find both homomorphisms : Z2 Z7 Z2 Z5 . How do you know that you've found them all? 6. For each of the following, find all the possible homomorphisms , and justify your results (Describe part (c) geometrically (rotate/reflect, etc.) rather than in cycle notation!). (a) : Z Z3 . : (b) Z3 Z. (c) : Z21 D39 . 7. Let G = ( G, ) be a group and let H be a subgroup of G. Fix an element a of G and let K = { aha-1 : h H }. (a) Prove that K is a subgroup of G. (b) Prove that the function : H K : h aha-1 is an isomorphism of groups. (You may quote basic properties of groups like the exponent laws and the cancellation properties.) 8. Prove directly that the centers of the following groups are as claimed: 1
(a) Z ( D4 ) = {0 , 2 }. (b) Z ( A4 ) = {e}. (c) Z (SL2 (R)) =
0 0
: = 1 .
What do the cosets of the center Z (SL2 (R)) look like in part (c)? 9. Either find explicitly an element g of the group G such that ghg-1 = k, of explain why no such element exists: (a) h = (123), k = (132) in S3 . (b) h = (1456)(23)(56), k = (1234)(56)(26) in S6 . (c) h = (1456)(23)(56), k = (12)(356) in S6 . 10. (Hard) Let X G be a fixed subset of a group G. Define the normalizer N ( X ) of X and centrali...